UDC 519.6 Computational models of diamond anvil cell compression A. I. Kondrat yev Independent Researcher, 5944 St. Alban Road, Pensacola, Florida 32503, USA Abstract. Diamond anvil cells (DAC) are extensively utilized in the study of material properties at extreme conditions of pressures and temperatures. The pressures in excess of 400 GPa and temperatures in excess of 4000 K have been reported in DAC devices. Most studies on the optimization of diamond geometry and gasket materials used in diamond anvil cell devices have been carried out by trial and error using experimental high pressure data. This article addresses design and optimization issues in DAC using finite element modeling (FEM) and computational analysis. The computational approach is centered around the recent advances in the growth of isotopically enriched carbon-13 layers on diamond anvil and their use as pressure sensor in diamond anvil cell devices. In particular, the carbon-13 diamond layer of 6 microns in thickness grown on top of existing anvil has been demonstrated to serve as a universal pressure sensor to 156 GPa. If a thin enough pressure sensor can be fabricated then the calibration of this sensor is independent of the stress or strain distribution in DAC. Some questions about the geometry and other properties of DAC remain open as well the theory of a universal pressure sensor. Keywords: Diamond anvil cell device, Finite-element modeling, Diamond-coated rhenium gasket, Diamond anvil failure conditions. 1. Introduction The object of our research is DAC device main component of which is diamond anvil. In DAC configuration a sample chamber is placed between the polished culets of diamond anvils. The metal gasket is preindented between the two anvils before drilling a hole that serves as a pressure chamber. All components are thoroughly assembled in a specially sealed cylinder. Then using tools we can apply pressure by turning special nuts in the assembly. Object of our investigation is the DAC device. We study separately DAC device components: diamond anvils, gasket, and sample material. In those components we study radial and axial stresses as well as pressure distribution with respect to geometry of anvils, sample and gasket material properties, and radial and axial coordinates. We have complete description of geometry and material properties of all DAC components. Because of DAC special design and diamond transparency we have X-ray diffraction experiments data for sample materials obtained within DAC compression. In some cases, diamonds were compressed to failure to test the upper limit of the pressures that can be created in diamond anvil cell E-mail: andykondratyev@aol.com.
device. In order to use results of diffraction experiments we have to use equations of state for diamond, sample material, and a gasket. As a result, we obtain dependencies of pressure distribution vs. radial distance. The pressure distribution in DAC can be experimentally measured only at the diamond/gasket interface and diamond/sample interface or at z = 0. Obtaining experimental data of pressure vs. axial distance is more complicated problem. At this stage of DAC device technology we are not able to measure axial and radial stresses. In order to calculate this information we have to use finite element analysis and modeling (FEAM) and computer simulation as well as analytical modeling and solution to main problems. Further development of finite element modeling (FEM) give us the full computer simulation model of DAC compression process. We develop and extend diamond anvil capabilities in order to use it as a high pressure sensor. When a thin film of 13 C layer was grown on a top of regular diamond anvil this thin film maybe used as a high pressure sensor using Raman spectroscopy. Our last study is by using analytical description and analysis of stresses and strains in an anvil and in a layer and also by using FEAM and computer simulation to analyze pressure distribution in a layer by describing the behavior of optical transverse phonon mode in an anvil and in a layer. Our working approach is based on use FEM and variety of analytical models with conjunction of experimental data. We validated these models as well computer simulation models on the results of other researchers and verify them on experimental data. The use of analytical model is essential because in some cases Nike2D supported simulation models do not provide this type of information, e.g. equations of state, dependence on initial and boundary conditions, solution stability, internal and external strain, stress, loads and phonoelastic tensor properties. Phonoelastic analytical models allow connect strain measurements with experimental data on Raman data under high pressure. 2. Computational Models. Equations of Motion and State in Nike2D Let ρ be the density, u i are the displacements Ω is the continuum domain, b i is body force per unit volume, τ ij is the Cauchy s stress tensor. As always u i = u i t, ü i = 2 u i t 2. FE equations of motion in Ω: ρ ü i = b i + τ ij,j. Let continuum domain Ω has a boundary Γ u, where spatial displacements u i are defined, and also a boundary Γ τ, where stresses τ are defined. u Boundary conditions are i = ũ i Γ u, τ ij n j = τ i Γ τ.
Initial conditions are u i(0) = u 0i, u i (0) = u 0i. Rate of deformation tensor is d ij = 1 2 ( u i,j + u j,i ). The Cauchy tensor τ in general is a function of d ij and a set of history variables H and temperature T (we consider temperature to be constant in our investigation) τ ij = τ ij ( d ij, H, T ) = τ ij ( d ij, H). In our work we use two Nike2D material models: elastic and elastoplastic. Nike2D uses axes r and z. Equations of state in the way they usually defined in solid mechanics are not inbuilt in Nike2D. Sets of parameters are used for elastic and elastoplastic materials. These parameters are related through some definitive equations of general form which we present later. This leads us in our research to a rather strong commitment: try to use well known equations of state for DAC components as well as different types of analytical models valid for FEM DAC (e.g. Timoshenko-Goodier s model). We use Lagrangian and Eulerian theories in order to implement
some of Nielsen s stress-strain models into FEAM. In order to validate analytical models we compare stress-strain curves obtained by Nike2D FEM with the ones obtained from Timoshenko-Goodier s, Nielsen s, and Hanfland et al. s models. We also develop adjusting procedures with use of some controlling parameters of these models in order to inbuilt them into FEAM. Notice that use of analytical models in our FEAM research is perfectly justified because these models are not inbuilt in FEM supported by Nike2D. Process of DAC compression is described by components, geometry model, material model, motion model and equation of state. Maze language commands allow to describe material models for each of DAC components. Diamond anvil and pusher are considered to be elastic and each of them is described by density, Young s modulus, and Poisson s ratio. This model describes isotropic, linear elastic material behavior. Gasket and sample material are considered to be kinematic/isotropic elastic-plastic material and described by density, Young s modulus, Poisson s ratio, yield stress, hardening modulus, hardening parameter, number of points in stress-effective plastic strain curve, effective plastic strain, and effective stress. This model includes linear or nonlinear strain hardening. Kinematic and isotropic hardening models which describe sample and gasket material yield identical behavior under monotonic loading. We consider only double-beveled diamond anvils. Using Nike2D Maze input language and using axial and radial symmetry of DAC we describe all four components of DAC. Each component is described by its own geometry and its own mesh. Geometry includes a series of line definitions. Points are introduced in the line definitions. Using these definitions element and nodal topologies are generated by Maze. Each DAC FE is defined by four boundary, nodal points which are in nondeformed state connected by line segments. The material number describing material properties of the element is also included in the definition. From lines parts and regions are constructed. Mesh assembly is defined from all described parts. Process of DAC compression is described by components, geometry model, material model, motion model and equation of state. Sample of DAC geometry and mesh for all components is shown. Maze language
commands allow to describe material models for each of DAC components. Diamond anvil and pusher are considered to be elastic and each of them is described by density, Young s modulus, and Poisson s ratio. This model describes isotropic, linear elastic material behavior. Gasket and sample material are considered to be kinematic/isotropic elasticplastic material and described by density, Young s modulus, Poisson s ratio, yield stress, hardening modulus, hardening parameter, number of points in stress-effective plastic strain curve, effective plastic strain, and effective stress. This model includes linear or nonlinear strain hardening. Kinematic and isotropic hardening models which describe sample and gasket material yield identical behavior under monotonic loading. Parameters describing elastic and elastoplastic material behavior usually are considered as linear or nonlinear functions of pressure. 3. Results and Conclusions The following results were obtained: 1. The validation of the modeling approaches. Pressure distribution results obtained on DAC compression model were verified using existing literature data; 2. DAC compression model in Nike2D computer code was investigated for different types of boundary and initial conditions, verified on several practical applications and theoretical results including equations of state, Timoshenko-Goodier s, Nielsen s and Hanfland et al. s models; 3. The experimentally measured pressure distribution in DAC to a peak pressure of 213 GPa by X-ray diffraction methods using a diamond coated rhenium gasket was compared with the finite element modeling results and a good agreement was obtained; 4. The role of a thin film of 13 C layer and its use as a universal sensor in DAC to 156 GPa and corresponding mechanisms involved were analyzed; 5. Finally, the role of diamond geometry and gasket materials were investigated to get a realistic estimate of the ultra high pressure conditions that can be generated in diamond anvil cells. The radial and axial stresses as well as shear stresses were examined and a failure criterion for diamond anvils in high pressure devices was developed. Acknowledgments This material is based upon work supported by the Department of Energy (DOE) National Nuclear Security Administration (NNSA) under Grant No DE-FG52-06NA26168.
References 1. Kondrat yev A. I., Vohra Y. K. Finite-element modeling of stresses and strains in a diamond anvil cell device: case of a diamond-coated rhenium gasket // High Pres. Res. 2007. Vol. 27, no. 3. P. 321 331. 2. Kondrat yev A. I. Mathematical modeling and computer simulation of compression process. VDM Verlag, 2009.